Download CT-Invariant Quantum Spin Hall Effect in Ferromagnetic Graphene

PRL 104, 066805 (2010)
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12 FEBRUARY 2010
PHYSICAL REVIEW LETTERS
CT-Invariant Quantum Spin Hall Effect in Ferromagnetic Graphene
Qing-feng Sun1,* and X. C. Xie1,2
1
2
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
(Received 13 July 2009; published 12 February 2010)
We predict a quantum spin Hall effect (QSHE) in ferromagnetic graphene under a magnetic field.
Unlike the previous QSHE, this QSHE appears in the absence of spin-orbit interaction and thus, is arrived
at from a different physical origin. The previous QSHE is protected by the time-reversal (T) invariance.
This new QSHE is protected by CT invariance, where C is the charge conjugation operation. Because of
this QSHE, the longitudinal resistance exhibits quantum plateaus. The plateau values are at 1=2, 1=6,
3=28, . . ., (in units of h=e2 ), depending on the filling factors of the spin-up and spin-down carriers. The
spin Hall resistance is also investigated and is found to be robust against the disorder.
DOI: 10.1103/PhysRevLett.104.066805
PACS numbers: 73.43.f, 72.25.b, 81.05.U
0031-9007=10=104(6)=066805(4)
are formed and the carriers move only along the edges. In
particular, the electrons (with their spins up) and holes
(with their spins down) move in opposite directions on a
given edge [see the inset in Fig. 1(a)]. Therefore, the QSHE
automatically exists in this system. Although the ordinary
metals (or doped semiconductors) cannot meet the above
two characteristics, a ferromagnetic graphene does.
Recently, several approaches to realize a ferromagnetic
graphene have been suggested [16–18]. For example, the
ferromagnetic graphene can be realized by growing the
graphene on a ferromagnetic insulator (e.g., EuO) [17]. For
a ferromagnetic graphene, as soon as the Fermi energy EF
is tuned to lie between the spin-up and spin-down Dirac
8
2
(a)
2
Ie/V (e /h)
6
1
4
3
M=0
M=0.01
M=0.02
M=0.03
2
4
0
2
-2
-4
3
1
4
-6
6
-8
5
(b)
Is/V (e/4π)
In the years since the spin Hall effect (SHE) has been
discovered, it has generated great interest [1–4]. In the
SHE, an applied longitudinal charge current or voltage
bias induces a transverse spin current due to the spindependent scatterings [1,2] of the spin-orbit interaction
(SOI) [3]. Soon afterwards, the quantum SHE (QSHE)
was also predicted [5,6]. The QSHE occurs in a topological
insulator in which the bulk material is an insulator with two
helical edge states carrying the current [7]. The edge states,
with opposite spins on a given edge or opposite edges for a
given spin direction containing opposite propagation directions, lead to a quantized spin Hall conductance. The
QSHE is a new quantum state of matter with a nontrivial
topological property. The existence of QSHE was first
proposed in a graphene film in which the SOI opened a
band gap and established the edge states [5,6]. But the
subsequent work found that the SOI in the graphene was
quite weak and the gap opening was small, so the QSHE
was difficult to observe [8]. Soon afterwards, the QSHE
was also predicted to exist in some other systems [9–12].
Recently, the QSHE was successfully realized in the
CdTe=HgTe=CdTe quantum wells, and a quantized longitudinal resistance plateau was experimentally observed
due to the QSHE [11].
Another subject that has also been extensively investigated in recent years is graphene, a single-layer hexagonal
lattice of carbon atoms [13], after it has been successfully
fabricated [14,15]. Graphene has a unique band structure
with a linear dispersion near the Fermi surface, giving it
many peculiar properties. For example, the quasiparticles
obey the Dirac-like equation and have relativisticlike behaviors, and its Hall plateaus are at the half-integer values.
In this Letter, we predict a new kind of QSHE in a
ferromagnetic graphene. Let us first imagine a twodimensional system consisting of the following characteristics: (i) its carriers contain electrons and holes; (ii) both
electrons and holes are completely spin-polarized with
opposite spin polarizations. When a high perpendicular
magnetic field is applied to the system, the edge states
2
EF
M=0
M=0.01
M=0.02
M=0.03
1
0
-0.2
-0.1
0.0
ε0
0.1
0.2
FIG. 1 (color online). The Hall conductance Ie =V (a) and spin
Hall conductance Is =V (b) vs. the energy 0 for N ¼ 80 and ¼
0:005. The two insets in (a) are the schematic diagram for the
four- and six-terminal graphene’s Hall bars. The inset in (b) is
the schematic diagram for band structure of the ferromagnetic
graphene while 0 þ M > EF > 0 M.
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Ó 2010 The American Physical Society
PRL 104, 066805 (2010)
PHYSICAL REVIEW LETTERS
points [see the inset in Fig. 1(b)], the above two characteristics are met and the QSHE occurs. In the following
calculations, we consider four- and six-terminal graphene
Hall bars (see the insets in Fig. 1(a)]. The results reveal that
the transverse spin current and spin Hall resistance indeed
show the quantized plateaus because of the QSHE.
Comparing this new QSHE with the previously-studied
QSHE, there are two essential differences: (i) The previous
QSHE comes from the SOI and the proposed systems all
contain the time-reversal symmetry [5–7,9,10], while the
present QSHE exists without the SOI and breaks the timereversal symmetry. However, this new QSHE is protected
by CT invariance. (ii) In the previous QSHE, the edge
states only carry a spin current while at equilibrium; in
this QSHE system, the edge states carry both spin and
charge currents at equilibrium with the two edges states
being CT partners of each other [see the inset in Fig. 1(a)].
Thus, this is a new kind of QSHE and the system is a new
type of topological insulator. Because of the topological
invariance, the plateaus of the spin Hall resistance are
robust to disorder or impurity scattering. So the plateau
is very stable and its value can be used as the standard
value for the spin Hall resistance.
In the tight-binding representation, the four- or sixterminal ferromagnetic graphene device [see the insets in
Fig. 1(a)] can be described by the Hamiltonian [19]
X
X
H ¼ ð0 MÞayi ai teiij ayi aj ; (1)
i;
hiji;
where ai and ayi are the annihilation and creation operators at the discrete site i. 0 is the on-site energy (i.e., the
Dirac-point energy), M is the ferromagnetic exchange split
[17], and t is the nearest neighbor hopping element. Here,
the whole device, including the center region and four or
six terminals, is made of the ferromagnetic graphene. With
the presence of a perpendicular magnetic field B, a phase
R
factor ij is added to the hopping element, ij ¼ ji A~ ~ 0 with the vector potential A~ ¼ ðBy; 0; 0Þ and 0 ¼
dl=
@=e.
The transmission coefficient Tpq ðÞ from the terminal
q to the terminal p with spin can be calculated from the
equation [20] Tpq ðÞ ¼ Tr½p Gr q Ga , where
p ðÞ ¼ i½rp ðÞ ap ðÞ, the Green functions
P r
cen
Gr ðÞ ¼ ½Ga ðÞy ¼ 1=½ Hcen
p p , and H
is the Hamiltonian of the center region. The retarded
self-energy rp ðÞ due to the coupling to the terminal p
can be calculated numerically [21]. After obtaining the
transmission coefficient, the particle current in the terminal
p with the spin can be calculated from
R the
P LandauerBüttiker
formula
Ip ¼ ð1=hÞ d q Tpq ðÞ where
fp ðÞ ¼ 1=fexp½ð ½fq ðÞ fp ðÞ,
p Þ=kB T þ 1g is the Fermi distribution function in the
terminal p, with the spin-dependent chemical potential
p and the temperature T. In the following numerical
calculations, we take t ¼ 1 as the energy unit and only
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consider the zero temperature case (T ¼ 0), as the thermal
energy kB T is normally much smaller than other energy
scales in the problem. The sample width is denoted by N,
and the insets of Fig. 1(a) show a system with N ¼ 3. In the
calculations, we choose N ¼ 80 and 40, and the corresponding widths are 33.9 and 16.9 nm.
pffiffiffi The magnetic field
is described by the with ð3 3=4Þa2 B=0 and the
magnetic flux in a honeycomb lattice is 2.
We first consider the four-terminal device (see the inset
at the top right corner of Fig. 1(a) and a small bias V is
applied between the longitudinal terminals 1 and 3 to study
the induced charge current Ine [Ine eðIn" þ In# Þ] and spin
current Ins [Ins ð@=2ÞðIn" In# Þ] in the transversal terminals 2 and 4. Here the boundary conditions for the four
terminals are 1" ¼ 1# ¼ eV=2, 2" ¼ 2# ¼ 0, 3" ¼
3# ¼ eV=2, and 4" ¼ 4# ¼ 0. The currents in the
terminals 2 and 4 satisfy the relations: I2e ¼ I4e Ie
and I2s ¼ I4s Is . Figures 1(a) and 1(b) show the Hall
conductance Ie =V and spin Hall conductance Is =V versus
the Dirac-point energy 0 , respectively. For a nonferromagnetic graphene (M ¼ 0) under the high magnetic field
( ¼ 0:005), Is =V is zero and Ie =V exhibits the plateaus at
odd integer values ne2 =h (n ¼ 1, 3, . . .) due to the
quantum Hall effect (QHE). These results have been observed in recent experiments [14,15]. For a ferromagnetic graphene with M Þ 0, the spin current emerges
[see Fig. 1(b)] since the QSHE. The spin Hall conductance
Is =V also shows the quantized plateaus. By considering the
edge state under the high magnetic field, the plateau values
of Is =V and Ie =V can be analytically derived to be at ð" # Þe=8 and ð" þ # Þe2 =2h [22], where is the Landau
filling factor for spin . In particular, when j0 j < jMj, in
which case the Fermi energy EF (EF ¼ 0) is located between the spin-up Dirac point 0 M and the spin-down
Dirac point 0 þ M, Ie is zero and a net quantum spin
current emerges in the transversal terminals. In addition, if
in the open circuit case, the spin accumulation emerges at
the sample boundaries instead of the spin current [22].
Since the QSHE can give rise to quantum plateaus in
resistances, we next study the longitudinal and Hall resistances in the six-terminal Hall device [see the inset in the
lower left corner of Fig. 1(a)]. Now we consider a small
bias V applied to the longitudinal terminals 1 and 4. The
transversal terminals 2, 3, 5, and 6 are all voltage probes,
their charge currents vanish (Ipe ¼ 0) and p" ¼ p# p . Combining these boundary conditions with the
Landauer-Büttiker formula, the voltages Vp (Vp ¼ p =e)
in four voltage probes can be obtained, then the longitudinal resistance R14;23 ¼ ðV2 V3 Þ=I14 and Hall resistance
R14;26 ¼ ðV2 V6 Þ=I14 are calculated, here I14 ¼ I1e ¼
I4e . The resistances contain the properties R14;26 ¼ R14;35
and R14;23 ¼ R14;65 .
Figures 2(a) and 2(b) show the longitudinal and Hall
resistances, R14;23 and R14;26 , versus the energy 0 and the
exchange split M at an external magnetic field ¼ 0:005.
Because of the QSHE and QHE, both R14;23 and R14;26 may
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PHYSICAL REVIEW LETTERS
(a)
0.5
2
R14,23 and R14,26 (h/e )
PRL 104, 066805 (2010)
0.4
0.3
0.2
0.1
0.0
-0.2
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4 (b)
-0.5
-0.2
-0.1
0.2
M=0.02
M=0.05
M=0.08
M=0.1
-0.1
0.0
0.1
ε0
M=0
M=0.02
M=0.05
M=0.08
0.0
ε0
0.1
0.2
FIG. 3 (color online). The resistances R14;23 (a) and R14;26 (b)
vs the energy 0 for N ¼ 80 and ¼ 0:005.
0.5
2
be nonzero, and they both exhibit plateau structures. The
plateau values are determined by the filling factors " and
# . For the fixed filling factors " and # , R14;23 and R14;26
maintain their plateau values regardless of 0 and M. By
considering the carriers transport along the edge states, the
plateau values can be analytically derived [22] R14;23 ¼ 0
and R14;26 ¼ ½1=ð" þ # Þh=e2 for ð" ; # Þ ¼ ðþ; þÞ or
(, ), and R14;23 ¼ ½j" # j=ðj" j3 þ j# j3 Þh=e2 and
R14;26 ¼ signð" Þ½ðj" j2 j# j2 Þ=ðj" j3 þ j# j3 Þh=e2 for
ð" ; # Þ ¼ ðþ; Þ or ( , þ). Some plateau values for
low " , # have been labeled in Fig. 2. The numerical
results in Fig. 2 are in excellent agreement with the analytic plateau values (the differences between them are less
than 106 ). Furthermore, R14;23 and R14;26 have the following properties: While j0 j > jMj with ð" ; # Þ ¼ ðþ; þÞ or
( , ), the longitudinal resistance R14;23 is zero and only
the Hall resistance R14;26 exists because the spin-up and
spin-down carriers are simultaneously either electronlike
or holelike and move in the same direction. On the other
hand, while j0 j < jMj with ð" ; # Þ ¼ ðþ; Þ or ( , þ),
the Fermi energy EF is located between 0 þ M and 0 M, the longitudinal resistance R14;23 emerges since now the
spin-up and spin-down carriers move in opposite directions
for a given edge. (i) While " ¼ # , the Hall resistance R14;26 ¼ 0, only the longitudinal resistance R14;23
exists with the value ð1=2Þh=e2 . This means that only
the QSHE emerges and the QHE vanishes in this region. In
this case, the system has the CT invariance. Furthermore, if
" ¼ # ¼ 1, R14;26 ¼ 0, and R14;23 ¼ ð1=2Þðh=e2 Þ.
Now the observed phenomena are completely the same
with the QSHE from the SOI [5–7,9,10], but their physical
Rs (h/e )
FIG. 2 (color). The panels (a) and (b) show the resistances
R14;23 and R14;26 (in the unit of h=e2 ) vs the exchange split M and
energy 0 for N ¼ 80 and ¼ 0:005.
mechanisms are different. (ii) While " Þ # but still
with ð" ; # Þ ¼ ðþ; Þ or ( , þ), R14;26 is now nonzero
since the numbers of the spin-up and spin-down edge states
are different. In this case, both resistances R14;26 and R14;23
have nonzero quantized plateaus and the QSHE and QHE
coexist. Figure 3 shows the resistances R14;23 and R14;26
versus the energy 0 for a fixed M (i.e., along the horizontal
lines in Fig. 2), and it clearly shows that the quantum
plateaus persist very well.
Up to now, we demonstrate the existence of QSHE in the
ferromagnetic graphene from both physical picture and
detailed numerical calculations. In the following, we study
the properties of the spin Hall resistance Rs , a measurable
quantity robust to dephasing [23]. and well reflecting the
topological invariance of the system. We again consider the
four-terminal Hall bar. But now the transversal terminals 2
and 4 are spin-biased probes with boundary conditions
Ip" ¼ Ip# ¼ 0 (p ¼ 2, 4). Here the spin Hall resistance
Rs is defined as the transversal spin bias over the longitudinal charge current: Rs ð2" 2# Þ=eI13 ¼
ð4" 4# Þ=eI13 . Since the spin bias n" n# is experimentally measurable, so is the Rs [24,25]. Figure 4
shows Rs versus the energy 0 for different ferromagnetic
exchange split M and magnetic field . For j0 j > jMj
with ð" ; # Þ ¼ ðþ; þÞ or ( , ), Rs ¼ 0. On the other
hand, while j0 j < jMj with ð" ; # Þ ¼ ðþ; Þ or ( , þ),
Rs exists. Rs exhibits the quantum plateaus, and its plateau
values are at ½1=ðj" j þ j# jÞh=e2 . For a small M (e.g.,
M ¼ 0:02t or 0:05t in Fig. 4(b)] or a high magnetic field [e.g., ¼ 0:005 in Fig. 4(a)], (" , # ) can only equal to
(a)
M=0.02
M=0.05
M=0.08
M=0.1
φ=0.005
φ=0.002
φ=0.001
0.4
0.3
0.2
0.1
(b)
0.0
-0.10
-0.05
0.00
ε0
0.05
0.10
-0.1
0.0
ε0
0.1
0.2
FIG. 4 (color online). (a) shows Rs vs 0 for M ¼ 0:05 and
(b) shows Rs vs 0 for ¼ 0:005. The parameter N ¼ 80.
066805-3
PRL 104, 066805 (2010)
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PHYSICAL REVIEW LETTERS
(a)
2
Rs (h/e )
ε0=0.02
0.3
ε0=0.04
ε0=0.06
0.2
0.1
ε0=0.08
W=0
W=1
W=2
W=3
0.0
-0.1
(b)
-0.2
-0.10
This work was financially supported by NSF-China
under Grants Nos. 10525418, 10734110, and 10821403,
China-973 program and US-DOE under Grants No. DEFG02- 04ER46124. Q. F. S. gratefully acknowledges
Professor R. B. Tao for many helpful discussions.
ε0=0
0.4
-0.05
0.00
ε0
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0.05
0
1
2
3
4
5
W
FIG. 5 (color online). Rs vs 0 (a) and Rs vs the disorder
strength W (b) with the parameters N ¼ 40, ¼ 0:007, and
M ¼ 0:05. The curves in (a) and (b) are averaged over up to
1000 and 8000 random configurations, respectively.
(1, 1), so only the plateau of Rs ¼ h=2e2 emerges. But
for a large M or a small magnetic field , (" , # ) may be
(1, 3), (3, 1), (1, 5), (5, 1), etc., then the plateaus of
Rs ¼ h=4e2 , h=6e2 , etc., are also possible.
Finally, we examine the disorder effect on the spin
Hall resistance Rs . Here we assume that the disorder only
exists in the central region [see dotted box in top right inset
of Fig. 1(a)]. Because of the disorder, the on-site energy
0 M for each site i in the central region is changed to
0 þ wi M, where wi is uniformly distributed in the
range [W=2, W=2] with the disorder strength W. Figure 5(a) shows Rs versus the energy 0 at the different
disorder strengths W and Fig. 5(b) shows Rs versus the
disorder strength W at different energies 0 . The results
show that the quantum plateaus of Rs are very robust
against the disorder because of the topological invariance
of the system. The quantum plateau maintains its quantized
value very well even when W reaches 2 [see Figs. 5(a) and
5(b)]. Since the plateau is so robust and stable, its value can
be used as the standard for the spin Hall resistance. In
addition, even for a very large disorder strength W (e.g.,
W ¼ 5 or larger), the plateau value only slightly decreases
while maintaining the plateau structure [see Fig. 5(b)].
This is because although the disorder strongly weakens
the spin bias 2" 2# , it also weakens the longitudinal
charge current I13 , so the value of Rs is affected less. This
means that in the large disorder limits (W ! 1), although
the QSHE is broken, the SHE still holds.
In summary, we predict a new QSHE in the ferromagnetic graphene film. Unlike the QSHEs studied so far, the
origin of this QSHE is not caused by the spin-orbit interaction. The results also show that the system can exhibit
the QSHE, the QHE, and the coexistence of the QSHE and
QHE, depending on the filling factors of the spin-up and
spin-down carriers. Because of the QSHE and QHE, both
the longitudinal and Hall resistances exhibit the plateau
structures. The plateau values (in the unit of h=e2 ) are at
1=2, 1=6, 3=28, . . ., for the longitudinal resistance and at
1=2, 1=4, 1=6, 2=7, . . ., for the Hall resistance. In
addition, the spin Hall resistance has also investigated and
found to be robust against the disorder.
*sunqf@aphy.iphy.ac.cn
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